Until recently, black holes (BHs) could be discovered only through accretion from other stars in X-ray binaries, or in merging double compact objects. Improvements in astrometric and spectroscopic measurements have made it possible to detect BHs also in non-interacting BH binaries (nBHB) through a precise analysis of the companions motion. In this study, using an updated version of the Startrack binary-star population modelling code and a detailed model of the Milky Way (MW) galaxy we calculate the expected number of detections for Gaia and LAMOST surveys. We develop a formalism to convolve the binary population synthesis output with a realistic stellar density distribution, star-formation history (SFH), and chemical evolution for the MW, which produces a probability distribution function of the predicted compact-binary population over the MW. This avoids the additional statistical uncertainty which is introduced by methods which Monte Carlo sample from binary population synthesis output to produce one potential specific realisation of the MW compact-binary distribution, and our method is also comparatively fast to such Monte Carlo realisations. Specifically, we predict $sim41$-$340$ nBHBs to be observed by Gaia, although the numbers may drop to $sim10$-$70$ if the recent ($lesssim100;$ Myr) star formation is low ($sim1;M_odot$/yr ). For LAMOST we predict $lesssim14$ detectable nBHBs, which is lower partially because its field-of-view covers just $sim6%$ of the Galaxy.
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